This invention relates to sonic fatigue testing methods for panel structures, and more particularly to a method of testing the response of a panel structure to acoustic pressure by simulating the sonic load spectrum of the acoustic pressure with a vibration spectrum on a shaker table.
Modern aircraft and missiles operate in environments wherein extreme acoustic pressures are common. The panel structures making up the aircraft must be able to withstand these extreme acoustic pressures. To ensure this ability, new panel structures for aircrafts are subjected to stringent acoustic testing.
Testing the response of a panel structure to acoustic pressure fluctuations, random both in time and space, essentially reduces to evaluating a dimensionless quantity known in the art as xe2x80x9cjoint acceptancexe2x80x9d. Joint acceptance corresponds to the coupling between the excitation pressure field and the structure. The joint acceptance function is defined by:             J      jm        ⁡          (      ω      )        =            1              A        2              ⁢                  ∫        a            ⁢                        ∫                      a            xe2x80x2                          ⁢                              C            ⁡                          (                                                r                  ∼                                ,                                                      r                    ∼                                    xe2x80x2                                ,                ω                            )                                ⁢                                    φ              j                        ⁡                          (                              r                ∼                            )                                ⁢                                    ϕ              m                        ⁡                          (                                                r                  ∼                                xe2x80x2                            )                                ⁢                      xe2x80x83                    ⁢                      ⅆ            a                    ⁢                      xe2x80x83                    ⁢                      ⅆ                          a              xe2x80x2                                          
dxc3xa3,dxc3xa3xe2x80x2=infinitesimal area vectors
C({tilde under (r)},{tilde under (r)}xe2x80x2, xcfx89)=cross-power spectral density coefficient of the sound pressure field
A=pressure surface area
{tilde under (r)},{tilde under (r)}xe2x80x2=space vectors locating points on the structure
The joint acceptance is computed by mapping the differential elements in integration space using the Jacobi method at the integrated degrees of freedom.
Displacement power spectral density response, Wy({tilde under (r)}),xcfx89) is equal to:                               W          y                ⁡                  (                      r            ∼                    )                    ,      ω        )    =                              A          2                ⁢                  G          ⁡                      (            ω            )                                      ω        4              ⁢                  ∑        j            ⁢                        ∑          m                ⁢                                                            φ                j                            ⁡                              (                                  r                  ∼                                )                                      ⁢                                          φ                m                            ⁡                              (                                  r                  ∼                                )                                      ⁢                                          J                jm                            ⁡                              (                ω                )                                                                        M              j                        ⁢                          M              m                        ⁢                          "LeftBracketingBar"                                                H                  j                                ⁡                                  (                  ω                  )                                            "RightBracketingBar"                        ⁢                          "LeftBracketingBar"                                                H                  m                                ⁡                                  (                  ω                  )                                            "RightBracketingBar"                                          
where             H      j        ⁡          (      ω      )        =      (                  -        1            +              2        ⁢                  ξ          j                ⁢                              ω            j                    ω                ⁢        Im            +                        ω          j          2                          ω          2                      )  
Mj, Mm=Jth, and mth elements of generalized mass matrix
xcfx86j(r)=jth normal mode shape
|xc2x7|=amplitude of complex variable
G(xcfx89)=reference power spectral density of sound pressure
Jjm(xcfx89)=joint acceptance
The root-mean-square displacement is given by       u    ⁡          (              r        ∼            )        =            (                        ∫                      ω            l                                ω            f                          ⁢                                            W              y                        ⁡                          (                                                r                  ∼                                ,                ω                            )                                ⁢                      xe2x80x83                    ⁢                      ⅆ            ω                              )              1      /      2      
Modal joint acceptance was first applied to a simply supported beam in xe2x80x9cOn The Fatigue Failure Of Structure Due To Vibrations Excited By Random Pressure Fieldsxe2x80x9d, Journal of Acoustical Society of America, Volume 30, No. 12, December 1958, Pages 1130-1135. Subsequent development yielded a method to determine the modal joint acceptance of a flat or cylindrical panel structure with arbitrary boundary conditions with a homogeneous random pressure field. Predictive methods were also developed to quantify the random excitation of the structure due to boundary layer noise.
Ideal testing of joint acceptance involves the manufacturing of a full sized prototype and then placing the prototype in an acoustic chamber wherein the sonic load spectrum of the acoustic pressures at a typical operating environment is duplicated. The response of each panel structure of the prototype is then monitored. As can be appreciated, however, building prototypes and subsequent testing in an acoustic chamber is impractical on a large scale basis due to the extreme complexity, time consumption and expense.
To avoid the complexity and expense of such test environments, most new panel structures are tested for acoustic response through simulation and analysis. Numerous prior art computational methods have been developed over the years to compute the behavior of such structural-acoustic systems. These modeling approaches can be separated into three main methods:
1. The acoustic-boundary integral method;
2. Finite element modeling representing the acoustic-structural interaction method involving fluid; and
3. The statistical energy-analysis method.
Of the above approaches, finite element modeling has found particular usefulness in the aircraft industry.
Finite element modeling involves the creation of a computer model of a proposed panel structure. The computer model is then tested against certain design criteria for its suitability. If the computer model is deemed suitable, a prototype panel is built and tested for its response to acoustic pressure by exposing it to a sonic load spectrum.
Unfortunately, such testing still requires the prototype panel to be subjected to a sonic load spectrum. This was deemed necessary since the response of a panel structure to a sonic load spectrum depends in a complex way on its position relative to the source, the existence of intervening structure, and the orientation of the panel in space. A significant drawback of such testing is the complexity and expense of generating the sonic load spectrum.
In view of the foregoing, it would be desirable to provide a method for testing a panel structure which obviates the need for generating the sonic load spectrum. For example, it would be advantageous to simulate the sonic load spectrum with an equivalent vibration spectrum. Such a vibration spectrum could easily be generated by, for example, an inexpensive shaker table.
In preferred embodiments, the present invention is directed towards a method of testing structural-acoustic systems using an inexpensive shaker test. In particular, the preferred embodiment of the present invention provides advantages over prior art techniques including simplified sonic-fatigue qualification testing, a fully parametric environment integrating structural-acoustic methods for computing the vibro-acoustic behavior of a structure under random excitation, the ability to capture large and complex geometries with arbitrary boundary conditions, and to investigate structural response for non-linear and thermal load based residual stress problems.
In one preferred embodiment, the method includes subjecting a model of a panel structure to computational acoustic and vibration loads. The method then computes the ratio of the maximum responses of the panel structure model to the acoustic and vibration loads. The ratio of these two maximum responses provides a conversion factor for linking an acoustic environment to a vibration environment. Using the conversion factor, a sonic load spectrum for testing against the panel structure is converted to a vibration load. The vibration load can then be applied to the panel structure using a shaker table.
To determine the conversion factor, it is presently preferred to apply two computational loads to a model of the panel structure. The first computational load consists of a 1 psi uniform pressure representing an acoustic pressure having a magnitude accounted for following a mode-superposition method. The second computational load consists of a 1 g negative base acceleration representing a vibration load having a response generated by a mode-acceleration method for random vibration.
Ideally, pressure load boundary conditions consistent with the in situ structure conditions of the panel structure are also applied to the model. Also, acceleration load boundary conditions consistent with the pressure load boundary conditions are preferably applied to the model of the panel structure. After applying the two computational loads, methodology determines the maximum pressure response of the model to the computational acoustic pressure load and the maximum acceleration response of the model to the computational acceleration load. The conversion factor is the ratio of the pressure response maximum to the acceleration response maximum.
In another preferred embodiment of the present invention, the above-determined conversion factor is used in sonic fatigue qualification testing of a panel structure. In this embodiment, the sonic pressure level that the panel structure is to be tested against is initially defined. The sonic pressure level is then converted to a pressure power spectral density using conventional techniques. The pressure power spectral density is then converted to an acceleration power spectral density using the conversion factor described above. The acceleration power spectral density is then used to generate input for a shaker table. The panel structure is then placed on the shaker table and monitored for its response to the input.